Laplacian Eigenvectors of Graphs

Perron-Frobenius and Faber-Krahn Type Theorems

  • Türker Biyikoğu
  • Josef Leydold
  • Peter F. Stadler

Part of the Lecture Notes in Mathematics book series (LNM, volume 1915)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. Pages 1-14
  3. Pages 15-27
  4. Back Matter
    Pages 93-115

About this book

Introduction

Eigenvectors of graph Laplacians have not, to date, been the subject of expository articles and thus they may seem a surprising topic for a book. The authors propose two motivations for this new LNM volume: (1) There are fascinating subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs. (2) "Geometric" properties of (cost) functions defined on the vertex sets of graphs are of practical interest for heuristic optimization algorithms. The observation that the cost functions of quite a few of the well-studied combinatorial optimization problems are eigenvectors of associated graph Laplacians has prompted the investigation of such eigenvectors.

The volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology.

Keywords

Eigenvector Graph Perron-Frobenius Theorem algorithms discrete Dirichlet problem graph Laplacian nodal domain vertices

Authors and affiliations

  • Türker Biyikoğu
    • 1
  • Josef Leydold
    • 2
  • Peter F. Stadler
    • 3
  1. 1.Işik UniversityIstanbulTurkey
  2. 2.Vienna University of Economics and Business AdministrationWienAustria
  3. 3.University of LeipzigLeipzigGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-73510-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2007
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-73509-0
  • Online ISBN 978-3-540-73510-6
  • Series Print ISSN 0075-8434
  • About this book